Problem: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-7z^2 - 77z - 168}{-9z^3 - 135z^2 - 504z}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-7(z^2 + 11z + 24)} {-9z(z^2 + 15z + 56)} $ $ x = \dfrac{7}{9z} \cdot \dfrac{z^2 + 11z + 24}{z^2 + 15z + 56} $ Next factor the numerator and denominator. $ x = \dfrac{7}{9z} \cdot \dfrac{(z + 8)(z + 3)}{(z + 8)(z + 7)}$ Assuming $z \neq -8$ , we can cancel the $z + 8$ $ x = \dfrac{7}{9z} \cdot \dfrac{z + 3}{z + 7}$ Therefore: $ x = \dfrac{ 7(z + 3)}{ 9z(z + 7)}$, $z \neq -8$